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General bounds for incremental maximization

Bernstein, Aaron ; Disser, Yann ; Groß, Martin ; Himburg, Sandra (2024)
General bounds for incremental maximization.
In: Mathematical Programming: Series A, Series B, 2022, 191 (2)
doi: 10.26083/tuprints-00023878
Article, Secondary publication, Publisher's Version

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Item Type: Article
Type of entry: Secondary publication
Title: General bounds for incremental maximization
Language: English
Date: 23 April 2024
Place of Publication: Darmstadt
Year of primary publication: February 2022
Place of primary publication: Berlin ; Heidelberg
Publisher: Springer
Journal or Publication Title: Mathematical Programming: Series A, Series B
Volume of the journal: 191
Issue Number: 2
DOI: 10.26083/tuprints-00023878
Corresponding Links:
Origin: Secondary publication DeepGreen

We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value k∈N that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all k between the incremental solution after k steps and an optimum solution of cardinality k. We consider a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general 2.618-competitive incremental algorithm for this class of problems, and we show that no algorithm can have competitive ratio below 2.18 in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be 1.58-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) d-dimensional matching, maximum (weighted) (b-)matching and a variant of the maximum flow problem. We show a general bound for the competitive ratio of the greedy algorithm on the class of problems that satisfy this relaxed submodularity condition. Our bound generalizes the (tight) bound of 1.58 slightly beyond sub-modular functions and yields a tight bound of 2.313 for maximum (weighted) (b-)matching. Our bound is also tight for a more general class of functions as the relevant parameter goes to infinity. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems, and our bounds for the greedy algorithm carry over both.

Uncontrolled Keywords: Incremental optimization, Maximization problems, Greedy algorithm, Competitive analysis, Cardinality constraint
Status: Publisher's Version
URN: urn:nbn:de:tuda-tuprints-238787
Additional Information:

Mathematics Subject Classification: 68W27 · 68W25 · 90C27 · 68Q25

Classification DDC: 500 Science and mathematics > 510 Mathematics
Divisions: 04 Department of Mathematics > Optimization
Date Deposited: 23 Apr 2024 12:45
Last Modified: 23 Apr 2024 12:46
SWORD Depositor: Deep Green
URI: https://tuprints.ulb.tu-darmstadt.de/id/eprint/23878
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